Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

Warm up and Homework Check

10 minutes

I include Warm upswith a Rubricas part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. This lesson’sWarm Up- Step Functions asks students to write a piecewise function given a scenario.

I also use this time to correct and record the previous day's Homework.

Warm Up- Step Functions

Warm up Rubric

Warm ups

Homework

Student Grouping

Introduction to Step Functions

15 minutes

This lesson begins with a simple step function scenario: AT&T offers a 10¢ per minute cell phone plan. I ask the students to both graph and write an equation for this situation (Math Practice 4). Most students will want to do f(x)=0.10x along with the corresponding graph. Once they have had some time to work, I put this situation on the board and tell them that there is a problem with this model. I then ask them to do a think-pair-share to figure out the problem. It is important to clarify at this point that talking for 2 minutes and 1 second will be counted as 3 minutes. I have them redraw this graph up through five minutes including the proper open and closed end points.

Now that they have a graph of this situation, I show how to write it as a floor function. We also talk about ceiling functions and the difference between the two.

I then give them another scenario to model with a function and a graph: The seniors are going on a field trip. Each bus has a capacity of 60 students. This will help solidify this new concept. I also have them find the domain, range and intercepts for this function. It is good to compare the graph for this specific scenario to the general graph of the function.

PowerPoint- Step Function

Is this a Step Function?

8 minutes

The next scenario isn’t really a step function although it does share similarities with one.

I took my dog to the groomer to get a bath. Here is the pricing: 0-30 pounds $15 31-70 pounds $25 > 70 pounds $35

I give this to my students without a warning that it is different. Once they have had a chance to work on it a bit, there is a good opportunity for some discussions about the similarities and differences between this and a step function (Math Practice 7). They will then write a piecewise function and graph this scenario.

Write your own Step Function

15 minutes

The last activity in this lesson begins with a brainstorming session on situations that can be modeled as step functions. Students will come up with examples as pairs and then we will make a list as a class. Each student will then create their own step function scenario which they will switch with their partner to graph and write a function (Math Practice 2). This activity will help students deepen their understanding of step functions vs. other piecewise functions.

Exit Ticket

2 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

Today’s Exit Ticket asks student to write a step function from a real life scenario.

Exit Ticket- Step Function

Homework

This short assignment gives the students 4 scenarios that are either step or piecewise functions. Students need to determine whether they are step or piecewise(Math Practice 7), write a function to match the scenario and graph it on a coordinate plane.